How restore the right normal-axis once for all faces?
This is only sort of possible. The normal of an object is kept when you separate it. However, even if you have a one-faced object, blender will not allow you to move the object based on its normals, because most objects have more than one face, which means the direction of motion would be conflicting.

You can move faces by their normal in edit mode though.

Simply change the transform orientation to "normal", go in edit mode Tab and you can transform by the normals.

In the image below, I started with a sphere, and separated out a face. I then went into edit mode on that face, and changed the transform orientation to that normal. Now I can move according to that normal.